Is there any solution to this quadratic Diophantine 3 variables equation?

Question

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation :

$$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$

Here $a, b, c$ are fixed positive integers.

Answer 1

One way to simplify a quadratic Diophantine is to get rid of cross-terms by doing a linear substitution. In this example, it is easy. Given,

$$x^2+2ax+y^2+2by = z^2+2cz\tag1$$

Let,

$$x,\,y,\,z = u-a,\,v-b,\,w-c$$

and it transforms to the simpler,

$$a^2+b^2-c^2-u^2-v^2+w^2=0$$

Since $a,b,c$ are constants, we can set $a^2+b^2-c^2 = d$, hence,

$$d-u^2-v^2+w^2=0$$

Do the further substitution $v,\,w =p+q,\,p-q$ to get,

$$d-4pq-u^2 = 0\tag2$$

I'm sure you can take it from here. P.S. All integer solutions of $(1)$ can be expressed in the form of $(2)$.